33,083 research outputs found
On the ADI method for the Sylvester Equation and the optimal- points
The ADI iteration is closely related to the rational Krylov projection
methods for constructing low rank approximations to the solution of Sylvester
equation. In this paper we show that the ADI and rational Krylov approximations
are in fact equivalent when a special choice of shifts are employed in both
methods. We will call these shifts pseudo H2-optimal shifts. These shifts are
also optimal in the sense that for the Lyapunov equation, they yield a residual
which is orthogonal to the rational Krylov projection subspace. Via several
examples, we show that the pseudo H2-optimal shifts consistently yield nearly
optimal low rank approximations to the solutions of the Lyapunov equations
Interpolatory methods for model reduction of multi-input/multi-output systems
We develop here a computationally effective approach for producing
high-quality -approximations to large scale linear
dynamical systems having multiple inputs and multiple outputs (MIMO). We extend
an approach for model reduction introduced by Flagg,
Beattie, and Gugercin for the single-input/single-output (SISO) setting, which
combined ideas originating in interpolatory -optimal model
reduction with complex Chebyshev approximation. Retaining this framework, our
approach to the MIMO problem has its principal computational cost dominated by
(sparse) linear solves, and so it can remain an effective strategy in many
large-scale settings. We are able to avoid computationally demanding
norm calculations that are normally required to monitor
progress within each optimization cycle through the use of "data-driven"
rational approximations that are built upon previously computed function
samples. Numerical examples are included that illustrate our approach. We
produce high fidelity reduced models having consistently better
performance than models produced via balanced truncation;
these models often are as good as (and occasionally better than) models
produced using optimal Hankel norm approximation as well. In all cases
considered, the method described here produces reduced models at far lower cost
than is possible with either balanced truncation or optimal Hankel norm
approximation
Inexact Solves in Interpolatory Model Reduction
We investigate the use of inexact solves for interpolatory model reduction
and consider associated perturbation effects on the underlying model reduction
problem. We give bounds on system perturbations induced by inexact solves and
relate this to termination criteria for iterative solution methods. We show
that when a Petrov-Galerkin framework is employed for the inexact solves, the
associated reduced order model is an exact interpolatory model for a nearby
full-order system; thus demonstrating backward stability. We also give evidence
that for \h2-optimal interpolation points, interpolatory model reduction is
robust with respect to perturbations due to inexact solves. Finally, we
demonstrate the effecitveness of direct use of inexact solves in optimal
approximation. The result is an effective model reduction
strategy that is applicable in realistically large-scale settings.Comment: 42 pages, 5 figure
- …